^{1}

^{2}

^{2}

^{1}

^{2}

The least-squares linear estimation problem using covariance information is addressed in discrete-time linear stochastic systems with bounded random observation delays which can lead to bounded packet dropouts. A recursive algorithm, including the computation of predictor, filter, and fixed-point smoother, is obtained by an innovation approach. The random delays are modeled by introducing some Bernoulli random variables with known distributions in the system description. The derivation of the proposed estimation algorithm does not require full knowledge of the state-space model generating the signal to be estimated, but only the delay probabilities and the covariance functions of the processes involved in the observation equation.

Originally, signal estimation problems were addressed under the assumption that the sensor data are transmitted over perfect communication channels, thus being received either instantaneously or with a known deterministic delay at the data processing center which provides the estimation. However, the use of communication networks for transmitting measured data motivates the need of considering eventual transmission delays and/or possible packet losses, due to numerous causes, such as random failures in the transmission mechanism, accidental loss of some measurements, or data inaccessibility at certain times. Often, these network uncertainties are random in nature and, hence, an appropriate model for such situations consists of describing the sensor delay or multiple packet dropout by a stochastic process, whose statistical properties are included in the system description. Therefore, estimation problems with bounded random delays in the observations and packet dropouts are challenging problems in networked control systems and have attracted much research interest.

Assuming that the state-space model of the signal to be estimated is known, many results have been reported on systems with random delays and packet dropouts. For example, Ray et al. [

For the packet-dropout problem, many recent papers can be mentioned. For example, the optimal

On the other hand, when the state-space model of the signal to be estimated is not available, it is necessary to use alternative information, for example, about the covariance functions of the processes involved in the observation equation. In this context, the least-squares (LS) linear and second-order polynomial estimation problems from randomly delayed observations based on covariance information have been addressed in [

In this paper, the LS linear estimation problem in systems with bounded random measurement delays and packet dropouts is addressed. The proposed estimators only depend on the delay probabilities at each sampling time, but do not need to know if a particular measurement is delayed or updated. Moreover, the estimation algorithm is derived using only covariance information. Consequently, considering the case of sensors with the same delay characteristics, the current study generalizes the results in [

The paper is organized as follows. In Section

In networked systems, such as telephone networks, cable TV networks, cellular networks or the Internet, among others, the system output is measured at every sampling time and the measurement is transmitted to a data processing center producing the signal estimation. In the transmission, delays and packet dropouts are unavoidable due to eventual communication failures and, to reduce the effect of such delays and packet dropouts without overloading the network traffic, each sensor measurement is transmitted for several consecutive sampling times, but only one measured output is processed for the estimation at each sampling time.

In this paper it is assumed that the largest delay is upper bounded (the bound is a known constant denoted by

In this section, the observation model with bounded random measurement delays and packet dropouts as well as the assumptions about the signal and noises involved, are presented.

Consider a signal vector,

The measured output

Therefore, the following model for the processed measurements to estimate the signal is considered:

This fact guarantees that if

Processed observations for

The signal estimation problem is addressed based on the following assumptions.

The

The noise process

For

Moreover,

For each

The estimation problem is addressed under the assumption that the evolution model of the signal need not be available and using only information about the covariance functions of the processes involved in the observation equation. Note that, although a state-space model can be generated from covariances, when only this kind of information is available, it is preferable to address the estimation problem directly using covariances, thus obviating the need for previous identification of the state-space model.

Although Assumption

Given the observation model (

The estimator

Since the innovations constitute a white process, this methodology allows us to find the orthogonal projection of the vector

Hence, to obtain the signal estimators it is necessary to find previously an explicit formula for the innovations and their covariance matrices.

The innovation at time

Similar that to (

From the model assumptions, it is clear that

To obtain the covariance matrix

Using this expression and the model assumptions, we have

In this section we obtain the state predictor (Section

From (

(i) On the one hand, using (

(ii) On the other hand, from (

Using (

Using (

Substituting into (

Substituting now (

Hence, an expression for

By substituting (

First, expression (

(i) Since

(ii) Since

Clearly, in view of (

Using (

A similar reasoning leads to the following expression:

Note that, from (

From the recursive relations (

The initial conditions of these equations, the prediction error covariance and cross-covariance matrices, are obtained as follows:

In fact, by writing

At the sampling time

Compute the innovation

Compute

Compute the filter

To implement the above steps at time

compute

compute the noise filter

for

Then, calculate the innovation

In this section, the application of the proposed signal estimation algorithm is illustrated by a simulation example. Consider a zero-mean scalar signal

Using the proposed filtering and fixed-point smoothing algorithms, we have estimated the signal from bounded random measurement delays and packet dropouts, assuming that the largest delay and the maximum number of successive dropouts is

As in the theoretical study, it is assumed that, at the initial time, the available measurement is equal to the real one,

If

Note that

For

updated (

delayed by one sampling period (

delayed by two sampling periods (

delayed by three sampling periods (

while

Next, the performance of the estimators is analyzed by investigating the error variances for

Filtering and smoothing error variances for

Signal

To analyze the performance of the proposed estimators versus the delay probabilities, the error variances have been calculated for different values of

Error variances versus

On the other hand, for each value of

Also, as expected, this improvement is more significant as

Next, we compare the filtering error variances at

Error variances versus

Error variances versus

In this paper, a recursive least-squares linear estimation algorithm is proposed to estimate signals from observations which can be randomly delayed or lost in transmission, a realistic and common assumption in networked control systems where, generally, transmission delays and packet losses are unavoidable due to the unreliable network characteristics. The largest delay and the maximum number of consecutive dropouts are assumed to be upper bounded by a known constant

Using an innovation approach, the estimation algorithm is derived without requiring the knowledge of the signal state-space model, but only the covariance functions of the processes involved in the observation equation, as well as the delay probabilities. To measure the performance of the estimators, the estimation-error covariance matrices are also calculated.

To illustrate the theoretical results established in this paper, a simulation example is presented, in which the proposed algorithm is applied to estimate a signal from bounded random measurement delays and packet dropouts, assuming that the largest delay and the maximum number of successive dropouts is

This research is supported by Ministerio de Educación y Ciencia (Grant no. MTM2008-05567) and Junta de Andalucía (Grant no. P07-FQM-02701).